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Let $K$ be a finite extension of $\mathbb{Q}_p$ and $L$ be a finite Galois extension of $K$. Let $V$ be a $p$-adic representation of $G_L$ Let $D$ be the $(\varphi, \Gamma)$-module associated to $V$ by Fontaine's functor. Is there an easy way to describe the $(\varphi, \Gamma)$-module associated to $Ind_{G_L}^{G_K} V$ in terms of $D$ ?

edited Dec 18 2010 at 12:56

asked Dec 18 2010 at 12:02

A M
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Laurent Berger · Accepted Answer · 2010-12-19 08:01:00Z UTC

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If $L/K$ is unramified, then the answer is exactly what Matt said. In the general case, you have to take into account the fact that $\Gamma_K$ is larger than $\Gamma_L$ and the construction is given in 2.2 of Ruochuan Liu's "Cohomology and Duality for (phi,Gamma)-modules over the Robba ring", see http://arxiv.org/abs/0711.4346

answered Dec 19 2010 at 8:01

Laurent Berger
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Thanks, it is exactly what I wanted to know. So basically, on the $(\varphi, \Gamma)$-module side, the construction is the "obvious" one. – A M Dec 20 2010 at 18:12

Emerton · Answer 2 · 2010-12-19 06:29:22Z UTC

I've always wondered this too. I think that if $L$ over $K$ is unramified, then we just take the $(\varphi,\Gamma)$-module for $V$, which is a $(\varphi,\Gamma)$-module over $\mathcal E_L$, and regard it as a $(\varphi,\Gamma)$-module over $\mathcal E_K$ (which is a subfield of $\mathcal E_L$). (Probably you already knew this, assuming it is correct.)

In the ramified case, I'm not sure; hopefully Laurent Berger will see this and answer!

Induced representations and $(\varphi, \Gamma)-modules

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